Expokit is a software package designed to compute matrix exponentials, including small matrix exponentials, the action of large sparse matrix exponentials on vectors, and solutions to systems of linear ODEs with constant inhomogeneity. It uses matrix-free Krylov subspace projection methods (Arnoldi and Lanczos processes) for sparse matrices, enabling efficient computation for large dimensions. The software supports real and complex matrices, with specific routines for symmetric and Hermitian matrices. It is particularly focused on computing transient states of Markov chains, which are critical in probabilistic modeling. The package addresses the numerical challenges of computing matrix exponentials, which are essential in various fields such as control theory, physics, and engineering. The computation of matrix exponentials is crucial for solving systems of ODEs and has applications in Markov chains, where the solution must satisfy probabilistic constraints. Expokit provides both full matrix exponentials and efficient methods for sparse matrices, using rational Chebyshev and Padé approximations for small matrices and Krylov subspace methods for large sparse matrices. The software includes routines for computing the matrix exponential times a vector, which is more efficient than computing the full matrix exponential. It also handles nonhomogeneous ODEs by incorporating the phi function, which allows for the solution of systems with constant forcing terms. Expokit is designed to be flexible and efficient, with support for both dense and sparse matrices, and it includes tools for error estimation and control, ensuring accurate and reliable results. The package is available through a web site and includes a mailing list for updates and user communication. It is implemented in Fortran 77 and uses BLAS and LAPACK for numerical computations, ensuring portability and efficiency. The software is self-contained, allowing it to operate even without the presence of these libraries. Expokit provides a range of routines for different matrix types and applications, making it a versatile tool for solving matrix exponential problems in various scientific and engineering contexts.Expokit is a software package designed to compute matrix exponentials, including small matrix exponentials, the action of large sparse matrix exponentials on vectors, and solutions to systems of linear ODEs with constant inhomogeneity. It uses matrix-free Krylov subspace projection methods (Arnoldi and Lanczos processes) for sparse matrices, enabling efficient computation for large dimensions. The software supports real and complex matrices, with specific routines for symmetric and Hermitian matrices. It is particularly focused on computing transient states of Markov chains, which are critical in probabilistic modeling. The package addresses the numerical challenges of computing matrix exponentials, which are essential in various fields such as control theory, physics, and engineering. The computation of matrix exponentials is crucial for solving systems of ODEs and has applications in Markov chains, where the solution must satisfy probabilistic constraints. Expokit provides both full matrix exponentials and efficient methods for sparse matrices, using rational Chebyshev and Padé approximations for small matrices and Krylov subspace methods for large sparse matrices. The software includes routines for computing the matrix exponential times a vector, which is more efficient than computing the full matrix exponential. It also handles nonhomogeneous ODEs by incorporating the phi function, which allows for the solution of systems with constant forcing terms. Expokit is designed to be flexible and efficient, with support for both dense and sparse matrices, and it includes tools for error estimation and control, ensuring accurate and reliable results. The package is available through a web site and includes a mailing list for updates and user communication. It is implemented in Fortran 77 and uses BLAS and LAPACK for numerical computations, ensuring portability and efficiency. The software is self-contained, allowing it to operate even without the presence of these libraries. Expokit provides a range of routines for different matrix types and applications, making it a versatile tool for solving matrix exponential problems in various scientific and engineering contexts.